Alhazen's problem

The medieval mathematician Ibn al-Haytham (Alhazen) solved an important problem known as Alhazen's problem in his work on catoptrics in Book V of the Book of Optics . The problem was first formulated by Ptolemy in 150 AD.^[1]

Geometric formulation

The problem comprises drawing lines from two points in a circle meeting at a third point on its circumference and making equal angles with the normal at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom off the edge of the table and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to an equation of the fourth degree.^[2][3][1]

Sums of powers

Ibn al-Haytham eventually derived a formula for the sum of fourth powers, where previously only the formulas for the sums of squares and cubes had been stated. His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in). He used his result on sums of integral powers to perform what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.^[4]

Influence

Ibn al-Haytham solved the problem using conic sections and a geometric proof, but later mathematicians such as Christiaan Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by complex numbers.^[5] An algebraic solution to the problem was finally found in 1997 by the Oxford mathematician Peter M. Neumann.^[6] Recently, Mitsubishi Electric Research Labs researchers solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors.^[7] They showed that the mirror reflection point can be computed by solving an eighth degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six.^[8] Alhazen's problem can also be extended to multiple refractions from a spherical ball. Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth degree equation.^[8]

References

Mathematics in medieval Islam
Mathematicians (Works)	9th century 'Abd al-Hamīd ibn Turk Sind ibn Ali Al-Abbās ibn Said al-Jawharī Al-Ḥajjāj ibn Yūsuf ibn Maṭar Al-Kindi De Gradibus Al-Mahani Banū Mūsā Hunayn ibn Ishaq al-Khwārizmī The Compendious Book on Calculation by Completion and Balancing ibn Qurra Na'im ibn Musa Sahl ibn Bishr Habash al-Hasib al-Marwazi 10th century Abd al-Rahman al-Sufi Book of Fixed Stars al-Būzjānī Abū Ja'far al-Khāzin Abū Kāmil Shujāʿ ibn Aslam Abu'l-Hasan al-Uqlidisi Abu-Mahmud Khojandi Ahmad ibn Yusuf Al-Nayrizi Al-Saghani Brethren of Purity Ibn Sahl Ibn Yunus Ibrahim ibn Sinan Muhammad ibn Jābir al-Harrānī al-Battānī Sinan ibn Thabit Al-Isfahani Nazif ibn Yumn Abū Sahl al-Qūhī 11th century Abū Ishāq Ibrāhīm al-Zarqālī Abu Nasr Mansur Abū Rayḥān al-Bīrūnī Ibn al-Haytham / Alhazen Book of Optics Ibn Muʿādh al-Jayyānī Al-Karaji Al-Sijzi Alī ibn Ahmad al-Nasawī Avicenna The Book of Healing Ibn Tahir al-Baghdadi Kushyar ibn Labban Yusuf al-Mu'taman ibn Hud 12th century Al-Khazini Ibn Yahyā al-Maghribī al-Samaw'al Omar Khayyám Jabir ibn Aflah Abu Bakr al-Hassar 13th century Muhyi al-Dīn al-Maghribī Nasīr al-Dīn al-Tūsī Zij-i Ilkhani Shams al-Dīn al-Samarqandī Sharaf al-Dīn al-Tūsī Ibn al‐Ha'im al‐Ishbili Ibn Abi al-Shukr 14th century Yaʿīsh ibn Ibrāhīm al-Umawī Ibn al-Banna' al-Marrakushi Ibn al-Shatir Kamāl al-Dīn Fārisī Al-Khalili Qotb al-Din Shirazi Ahmad al-Qalqashandi 15th century Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī Ali Qushji Jamshīd al-Kāshī Qāḍī Zāda al-Rūmī Ulugh Beg Ibn al-Majdi 16th century Al-Birjandi Muhammad Baqir Yazdi Taqi al-Din Ibn Hamza al-Maghribi Ibn Ghazi al-Miknasi
Other treatises	Almanac Encyclopedia of the Brethren of Purity Tables of Toledo Tabula Rogeriana Zij Zij-i-Sultani
Concepts	Alhazen's problem Islamic geometric patterns
Centers	Al-Azhar University Al-Mustansiriya University House of Knowledge House of Wisdom Istanbul observatory of Taqi al-Din Madrasah Maktab Maragheh observatory University of Al-Karaouine
Influences	Babylonian mathematics Greek mathematics Indian mathematics
Influenced	Byzantine mathematics European mathematics Indian mathematics